Rinse greens well.
Cut chard leaves into thin shreds; thinly slice stems.
(Or cut and discard tough stems from kale; cut leaves into thin shreds.) You should have about 8 cups, lightly packed.
Set greens aside.
Place a wok or 12- to 14-inch frying pan over high heat.
Add oil.
Crumble sausage into pan.
Stir frequently until meat is well browned.
Add onions, garlic, mushrooms, nutmeg, pepper, and oregano; cook, stirring often, until all liquid has evaporated and onions are soft.
Stir in greens a portion at a time, starting with stems, until all greens are in pan and just wilted, 3 to 4 minutes.
Add beaten eggs to pan.
Stir over low heat just until softly set.
Season to taste with salt.
Transfer to a serving dish and sprinkle with cheese.